### Elegant Algebraic Expressions for Finding Surface Area

Saturday, October 17, 2009
I always wonder why we use those "elegant" algebraic expressions in math class. Well, I could probably answer my own questions. We use those formulas because, it's easier to understand (I think..) and when we write it out, it looks elegant, or beautiful, or whatever. I also think that we use those formulas because, mathematicians use it. We want to be like them!!!! :D

So, I'm here to blog about what formulas/ algebraic expressions are elegant for different kinds of shapes. Also to give examples on how to use them.

What is the most elegant algebraic expression...

for finding the surface area of a cube?

-A cube simply has 6 faces that are equal, so whatever length on one side is squared.

To find the surface area of a cube, you simply use this formula:

S.A.= 6s²

Just for an example on how to use that formula, I will find the surface area of the cube with a side length of 5.

S.A.= 6s²

= 6 (5²)

= 6 (25)

S.A.=150 u²

What is the most algebraic expression...

for finding the surface area of a rectangular prism?

-We already learned this last year, so I think you're familiar with the formula. A rectangular prism has 3 sets of 2 equal faces (each set), that equals to 6 faces in total.

In finding the surface area of a rectangular prism, you could use this formula:

S.A.+ 2(lw)+2 (hw)+ 2 (hl)

For example on how to use that formula for finding the surface area of a rectangular prism, I will use it, using the measurements shown above (in the picture).

S.A.= 2 (lw)+ 2 (hw)+ 2 (hl)

= 2 (7) (4) + 2 (10) (4) +2 (10) (7)

= 2 (28) + 2 (40) + 2 (70)

=56+80+140

S.A. = 276 cm²

* NEVER forget to put the unit (eg. cm)/ measurement and the squared sign (²) in your answers.

What is the most elegant algebraic expression...

for finding the surface area of a cylinder?

-We already learned this last year too. So, you're most likely to know this formula. You have to find the surface area of the 2 circles, then add the rectangle. The find the surface area of the rectangle, you multiple the height by the circumference of the circle.

To find the surface area of a cylinder, you use this formula:

S.A. = 2πr²+ 2πrh

I'm going to use the measurements above, in giving an example on how to use the formula.

S.A.=2πr²+2πrh

=2(3.14)(5²)+2(3.14)(5)(15)

= 2 (3.14) (25)+ 2 (3.14) (75)

= 2 (78.54)+2 (235.62)

= 157.08+ 471.24

S.A.= 628.32cm²

What is the most elegant algebraic expression...

in finding the surface area of a triangular prism?

-Triangular prisms also uses formulas. But, it's kind of mixed. You have to use the Pythagorean theorem, the formula for finding the area of the triangles, and finally, the formula for finding the area of a rectangle.

Area of 2 triangles= bh

= (10)(15)

= 150

Area of base= lw

= (10)(20)

= 200

*For finding the area of the roofs, most of the time, only one side is given. So you use the Pythagorean theorem to find the hypotenuse of the triangle, which is the unknown side. Then you multiply that by the side that is given (width).

a²+b²= c²

10²+15²=c²

100+225=c²

325=c²

√325=√c²

18.03=c

Area of 2 rectangles (roofs) = 2(lw)

= 2(18.03)(20)

= 2(360.55)

=721.11

Then, you add all of the areas together, to get the total surface area.

S.A.=Area of triangles+Area of base+Area of roofs

=150+200+721.11

=1071.11

What is the most elegant algebraic expression...

in finding the surface area of a composite object?

- There's no formula to use in particular. It all depends on what shapes are combined.

In this particular shape, I would use the formula for finding the surface area of a rectangular prism. I would have to use that twice (top and bottom). Add them together. Then, find the area of the overlap, then subtract.

S.A. (bottom)= 2(lw)+2 (hw)+ 2(hl)

= 2 (5)(2) + 2 (2)(2)+ 2 (2)(5)

= 2 (10)+2 (4)+ 2 (10)

= 20+8+20

= 48

S.A. (top)= 2(lw)+2(hw)+2(hl)

= 2(5)(1)+2 (2)(1)+2(2)(5)

= 2(5)+2(2)+2(10)

= 10+ 4+ 20

= 34

Area of overlap = (1)(5)(1)

= 5

S.A.=area (bottom)+ area (top) -area of overlap

=48+34-5

S.A. =77 u²

How does symmetry help us solve some of these surface area problems?

- Well, that's an interesting question. It's also an easy question to answer. Most solid shapes (eg. rectangular prism) has planes of symmetry in it. Planes of symmetry helps us how many faces are equal. Which helps us to know that when we see one side of that shape, the opposite side is the same. Planes of symmetry only applies to some shapes. So, it can't be like that for every shape.

What happens when a part of any of these shapes is missing? How do I find the surface area then?

-Simple! Most shapes have planes of symmetry. So if one side is missing, you will know that it's the same as the given part. That helps on finding the missing part of a shape, then calculating the surface area with the missing shape.

Hey! Feel free to comment on how great this post is! Also feel free to criticize this post! Or just comment! :D

#### 3 comments:

kara 9-05 said...

Awesome JOB ANGELA!! This post is great! You explained everything in detail, the pictures are huge and easy to see, and you have all of the formulas! The only thing that I would suggest is that maybe you could double check some of your grammer, I don't even think that there is anything wrong with your grammer, but it wouldn't hurt to check. Anyways. GOOD JOB! (it really stands out)

melanie905 said...

Great post Angela! I liked how you used a lot of colours, and your examples and explanations are very clear. Nice pictures, and again, AWESOME POST!

Lissa 9-05 said...

Awesome post Angela! Your colours are really nice and I like your pictures (: